Posted tagged ‘Creator’

This is the seventh of eleven Letters of Euler I will rewrite and post on the subject of infinitesimals (the infinitely small), an idea that is fundamental to a good understanding of calculus. Click here to read the previous letter.

Principle of the satisfying Reason, the strongest Support of the Monadists.

You must be perfectly sensible that one of the two systems, which have undergone such ample discussion, is necessarily true, and the other false, seeing they are contradictory. It is admitted on both sides, that bodies are divisible: the only question is, “Whether this divisibility is limited?”, or “Whether it may always be carried further, without the possibility of ever arriving at indivisible particles?”

The system of monads is established in the former case, since after having divided a body into indivisible particles, these very particles are monads, and there would be reason for saying that all bodies are composed of them, and each of a certain determinate number. Whoever denies the system of monads, must likewise, then, deny that the divisibility of bodies is limited. he is under the necessity of maintaining, that it is always possible to carry this divisibility further, without ever being obliged to stop; and this is the case of divisibility in infinitum, on which system we absolutely deny the existence of ultimate particles: consequently the difficulties resulting from their infinite number fall to the ground themselves. In denying monads, it is impossible to talk any longer of ultimate particles, and still less of the number of them which enters into the composition of each body.

You must have remarked, that what I have hitherto produced in support of the system of monads is destitute of solidity. I now proceed to inform you that its supporters rest their cause chiefly on the great principle of the sufficient reason, which they know how to employ so dexterously, that by means of it they are in a condition to demonstrate whatever suits their purpose, and to demolish whatever makes against them. The blessed discovery made, then, is this, “That nothing can be without a sufficient reason;” and to modern philosophers we stand indebted for it.

In order to give you an idea of this principle, you have only to consider, that in every thing presented to you, it may always be asked, “Why it is such?” And the answer is what they call the sufficient reason, supporting it really to correspond with the question proposed. Wherever the “why” can take place, the possibility of a satisfactory answer is taken for granted, which shall, of course, contain the sufficient reason of the thing.

This is very far, however, from being a mystery of modern discovery. Men in every age have asked “why;” an incontestable proof of their conviction that every thing must have a satisfying reason of its existence. This principle, that nothing is without a cause, was very well known to ancient philosophers; but unhappily this cause is for the most part concealed from us. To little purpose do we ask “why:” no one is qualified to assign the reason. It is not a matter of doubt, that every thing has its cause; but a progress thus far hardly deserves the name; and so long as it remains concealed, we have not advanced a single step in real knowledge.

You may perhaps imagine, that modern philosophers, who make such a boast of the principle of a satisfying reason, have actually discovered that of all things, and are in a condition to answer every why that can be proposed to them; which would undoubtedly be their very summit of human knowledge; but, in this respect, they are just as ignorant as their neighbors: their whole merit amounts to no more than pretension to have demonstrated, that wherever it is possible to ask the question “why,” there must be a satisfying answer to it, though concealed from us.

They readily admit, that the ancients had a knowledge of this principle, but a knowledge very obscure; whereas they pretend to have placed it in its clearest light, and to have demonstrated the truth of it: and therefore it is that they know how to turn it most to their account, and that this principle puts them in a condition to prove, that bodies are composed of monads.

Bodies, they say, must have their sufficient reason somewhere, but if they were divisible to infinity, such reason could not take place: and hence they conclude, with an air altogether philosophic, “that, as every thing must have its sufficient reason, it is absolutely necessary that all bodies should be composed of monads:” which was to be demonstrated. This, I must admit, is a demonstration to be resisted.

It were greatly to be wished that a reasoning so slight could elucidate to us questions of this importance; but I frankly confess, I comprehend nothing of the matter. They talk of the sufficient reason of bodies, by which they mean to reply to a certain “wherefore,” which remains unexplained. But it would be proper, undoubtedly, clearly to understand, and carefully to examine a question, before a reply is attempted; in the present case, the answer is given before the question is formed.

Is it asked, “Why do bodies exist?” It would be ridiculous, in my opinion, to reply, “Because they are composed of monads;” as if they contained the cause of that existence. Monads have not created bodies: and when I ask, “Why such a being exists?” I see no other reason that can be given but this, “Because the Creator has given it existence;” and as to the manner in which creation is performed, philosophers, I think, would do well honestly to acknowledge their ignorance.

But they maintain, that God could not have produced bodies, without having created monads, which were necessary to form the composition of them. This manifestly supposes, that bodies are composed of monads, the point which they meant to prove by this reasoning. And you are abundantly sensible, that it is not fair reasoning to take for granted the truth of a proposition which you are bound to prove by reasoning. It is a sophism known in logic by the name of a petitio principii, or, begging the question.

This is the fifth of eleven Letters of Euler I will rewrite and post on the subject of infinitesimals (the infinitely small), an idea that is fundamental to a good understanding of calculus. Click here to read the previous letter.

Letter XI. Reflections on Divisibility in infinitum, and on Monads.

In speaking of the divisibility of body, we must carefully distinguish what is in our power, from what is possible in itself. In the first sense, it cannot be denied, that such a division of a body as we are capable of, must be very limited.

By pounding a stone we can easily reduce it to powder; and if it were possible to reckon all the little grains which form that powder, their number would undoubtedly be so great, that it would be matter of surprize, to have divided the stone into so many parts. But these very grains will be almost indivisible with respect to us, as no instrument we could employ would be able to lay hold of them. But it cannot with truth be affirmed that they are indivisible in themselves. You have only to view them with a good microscope, and each will appear itself a considerable stone, on which are distinguishable a great many points and inequalities; which demonstrates the possibility of a farther division, though we are not in a condition to execute it. For wherever we can distinguish several points in any object, it must be divisible into so many parts.

We speak not, therefore, of a division practicable by our strength and skill, but of that which is possible in itself, and which the Divine Omnipotence is able to accomplish.

It is in this sense, accordingly, that philosophers use the word ‘divisibility:’ so that if there were a stone so hard that no force could break it, it might be without hesitation affirmed as divisible, in its own nature, as the most brittle, of the same magnitude. And how many bodies are there on which we cannot lay any hold, and of whose divisibility we can entertain not the smallest doubt? No one doubts that the moon is a divisible body, though he is incapable of detaching the smallest particle from it: and the simple reason for its divisibility, is its being extended.

Wherever we remark extension, we are under the necessity of acknowledging divisibility, so that divisibility is an inseparable property of extension. But experience likewise demonstrates that the division of bodies extends very far. I shall not insist at great length on the instance usually produced of a ducat*: the artisan can beat it out into a leaf so fine, as to cover a very large surface, and the ducat may be divided into as many parts as that surface is capable of being divided. Our own body furnishes an example much more surprizing. Only consider the delicate veins and nerves with which it is filled, and the fluids which circulate through them. The subtility there discoverable far surpasses imagination.

*A ducat is a gold coin used in Euler’s day.

The smallest insects, such as are scarcely visible to the naked eye, have all their members, and legs on which they walk with amazing velocity. Hence we see that each limb has its muscles composed of a great number of fibres; that they have veins, and nerves, and a fluid still much more subtile which flows through their whole extent.

On viewing with a good microscope a single drop of water, it has the appearance of a sea; we see thousands of living creatures swimming in it, each of which is necessarily composed of an infinite number of muscular and nervous fibres, whose marvellous structure ought to excite our admiration. And though these creatures may perhaps be the smallest which we are capable of discovering by the help of the microsope, undoubtedly they are not the smallest which the Creator has produced. Animacules probably exist as small relatively to them, as they are relatively to us. And these after all are not yet the smallest, but may be followed by an infinity of new classes, each of which contains creatures incomparably smaller than those of the preceding class.

We ought in this to acknowledge the omnipotence and infinite wisdom of the Creator, as in objects of the greatest magnitude. it appears to me, that the consideration of these minute species, each of which is followed by another inconceivably more minute, ought to make the liveliest impression on our minds, and inspire us with the most sublime ideas of the works of the Almighty, whose power knows no bounds, whether as to great objects or small.

To imagine that after having divided a body into a great number of parts, we arrive, at length, at particles so small as to defy all farther division, is therefore the indication of a very contracted mind. But supposing it is possible to descend to particles so minute as to be, in their own nature, no longer divisible, as in the case of the supposed monads; before coming to this point, we shall have a particle composed of only two monads, and this particle will be of a certain magnitude or extension, otherwise it could not have been divisible into these two monads. Let us farther suppose, that this particle, as it has some extension, may be the thousandth part of an inch, or still smaller if you will; for it is of no importance, what I say of the thousandth part of an inch may be said with equal truth of every smaller part. This thousandth part of an inch, then, is composed of two monads, and consequently two monads together would be the thousandth part of an inch, and two thousand times nothing, a whole inch; the absurdity strikes at first light.

The partisans of the system of monads accordingly shrink from the force of this argument, and are reduced to a terrible nonplus when asked how many monads are requisite to constitute an extension. Two, they apprehend, would appear insufficient, they therefore allow that more must be necessary. But, if two monads cannot constitute extension, as each of the two has none; neither three, nor four, nor any number whatever will produce it; and this complexity subverts the system of monads.

The following is the third in a series of posts covering Shormann Mathematics, Algebra 1, the newest product from DIVE Math and Science! Click here to read the complete document that covers Shormann Math core ideas, course description, and Algebra 1 table of contents.

Jesus Christ is the “Common Core” of Shormann Math

Perhaps you have heard of the United States government’s “Common Core” curriculum. Perhaps you have also heard that a lot of people are concerned about it. Leading experts believe the Common Core’s mathematics standards will not prepare students to study science, technology, engineering, and math (STEM) in a selective four-year college. And a white paper by the Pioneer Institute concludes by saying

“At this time we can conclude only that a gigantic fraud has been perpetrated on this country, in particular on parents in this country, by those developing, promoting, or endorsing Common Core’s standards.”

Unfortunately, man and his ever-changing ideas are at the core of this curriculum. At DIVE, we strive to place Jesus Christ at the core of all our products, and we pray that this will result in students learning math and science for His glory and the service of others. So, even though our primary goal is NOT to prepare students for STEM, we believe by putting Christ at the foundation, just like the world’s original universities did, students will naturally learn to use mathematical tools that will connect them to their world and their Creator.

Belief in evolutionism requires one to reject the authority of Scripture regarding special creation of humans, along with different created kinds, or baramins, of living organisms. Biblical history must also be rejected, because millions of years are apparently required for nature to perform its evolutionary magic. Belief in evolutionism forces one to cling to a number of 19th and 20th Century hypotheses that use artificial constructs like the geologic column and Thomas Malthus’ population data as evidence for bacteria-to-people evolution.

Fortunately, the more we learn about Earth and life in the 21st Century, the more they proclaim the glory of their Creator and the Truth of the incredible story revealed in Scripture. An example of 21st Century research is the mounting evidence against the idea of molecular clocks. Natural history researchers look at differences in genes along with fossil evidence to determine when two species diverged from a common ancestor. For the human species, researchers use molecular clocks to predict the date of “Mitochondrial Eve”, our Most Recent Common Ancestor (MRCA) that supposedly originated in Africa.

Molecular clocks came into use in the 1960s. In the 1990 edition of Biology by Neil Campbell, an age between 200,000 and 400,000 years is given for “Eve” (p. 669). Moving ahead to 2004, we find in the 10th edition of Biology by Starr and Taggart that Eve is now only 100,000 to 200,000 years old (p. 471). The fact that the estimates were cut in half, on top of the huge error involved (50%), would make any reasonable scientist question molecular clocks.

And they do. As we entered the 21st century, we saw F.J. Ayala’s paper titled “Molecular Clock Mirages”. In 2006, world renowned evolutionary biologist Thomas Cavalier-Smith stated in a paper “Evolution is not evenly-paced and there are no real molecular clocks.” And then there’s F. Chang’s study using genealogy and statistics to predict an MRCA of less than 1,000 years ago. Chang began with an overly-simplified model, so over the next few years he added to it, and in 2003 colleague D. Rohde published research revealing an MRCA of between 2,000 and 5,000 years ago. And molecular clock skeptics Thorne and Wolpoff voiced their opinions in the 2003 Human Evolution Special Issue of Scientific American, flatly stating “putting aside the idea of a molecular clock, one can interpret the genetic data in a much more reasonable way.” (p. 52).

In 2004, Rohde, Chang, and Olson published their latest findings in Nature, and their computations shift the MRCA from Africa to somewhere in Asia. They also calculated that “all modern individuals have identical ancestors by about 3,000 BC.” Mentioning that their computer simulation was “far too conservative”, they used some more realistic numbers to come up with a “mean MRCA date is as recent as AD 55 and the mean IA date is 2,158 BC.”

The identical ancestors (IA) point differs from the MRCA. The MRCA is believed to have had many contemporaries of both sexes, and some of these also left unbroken chains of descendents down to today’s population. The IA differs in that it pushes further back in time to the point where populations can be divided into two groups: a group that left no descendents today, and a group from which all modern humans descended from. Such a scenario could arise from a population bottleneck, and the obvious example that comes to mind is the Flood described in the book of Genesis, which occurred around 2500 BC. The date of the Flood is within the range of IA dates computed by Rohde et al. During the Flood, a human population of 8 survived, and all others perished. While Rohde et al’s research does not “prove” the Genesis Flood, it definitely doesn’t rule it out.

In 2008, a paper by Matsen and Evans tried to tie genetics with the genealogy of Rohde et al, and they simply concluded genetic diversity is related to the number of descendants, confirming the ability of Rohde et al’s model to explain the human diversity we see today as resulting from a very recent ancestor.

21st Century research using genealogies instead of genetics may be a bit confusing, but the reason for that is not just the complex mathematics involved, but the basic fact that confusion exists over what happened in the past. To add to the confusion, in 2008 fossil collectors discovered a human footprint alongside that of a dinosaur. Fossil evidence like this is no problem for Christians who trust the historical accuracy of Scripture, but it is a huge contradiction to many others’ beliefs about the past. Is this fossil real?

The truth is, there will ALWAYS be confusion about what happened in the past because we cannot go back and verify it. Natural history research is not the same thing as testable, repeatable science, and should be approached as a “mixed question”, requiring inputs not just from science, but mainly history, followed by art, and philosophy. “Belief” in past events though is based on faith, either a God-given faith (Ephesians 2:8-9)in the story revealed in Scripture, or a blind faith in something else. Everybody believes in something, what do you believe?

Crying Wolf is an excellent documentary by a homeschool graduate, Jeffrey King, about properly understanding man’s role in nature. It is available to watch free online until December 2011. It does contain some pretty graphic images of animals destroyed by wolves and left to die, but he warns you before the images appear. As one man says in the movie, environmentalism is nature worship, and as Christians, we are supposed to worship the Creator, not the creature (Romans 1:24-25). We are supposed to take what God has given us and make it fluorish. We are supposed to be a part of nature, not just wimpy spectators who feel bad about our “carbon footprint”. Watch the movie and add a comment below if you want to discuss it. Then, get outside and grow something, shoot something, catch something, cut down a tree, etc., but do it responsibly and make good use out of it.

“spam” is a word defining “unsolicited electronic advertising”. It is a form of advertising most people consider as underhanded and an abuse of the Internet. When received on a blog, it is also referred to as “blam”. Last week, I received a “spam alert” on my blog. This is not the first spam comment I’ve received, but this one surprised me because it was from an abortion clinic:

abortion clinic spam message

Before I continue, allow me to make it crystal clear how myself and millions (billions?) of others define abortion. Abortion is murder. Here is a drawing detailing the moment a new life begins to grow:

A new baby begins growing the moment the egg is fertilized.

A new baby begins growing the moment the egg is fertilized. Unfortunately, many people don’t agree or are ignorant of this fact, and in America and most other countries, killing unborn children, a.k.a “abortion”, is actually legal. However, when an unborn baby is taken from its mother and killed, only a fool would define that act as something other than murder.

So why would a child-killing clinic send out spam messages? Well, mainly because the killing of unborn babies is a multi-million dollar industry in the United States and elsewhere. It is filled with greedy people who are out to make a dollar, and they have no problem using unethical practices like “spamming” to advertise their immoral businesses. Normal advertising requires a payment to someone for marketing your product. Spammers basically eliminate the middle-man, sending out large volumes of unsolicited advertising, greatly reducing their advertising costs. Spamming is selfish, greedy and wrong, but what else would you expect from a company who tries to deceive women into believing that killing their unborn child is not murder?

According to a Wikipedia page on blog spam, a company benefits from sending out blog spam because “Adding links that point to the spammer’s web site artificially increases the site’s search engine ranking. An increased ranking often results in the spammer’s commercial site being listed ahead of other sites for certain searches, increasing the number of potential visitors and paying customers.” Do you see the connection? The higher the search engine ranking is for a child-killing operation like the Ocala Women’s Center in Florida, the more children they will get paid to kill.

The blog spam I received was for the Legal Abortion by Pill Clinic, operated by the Ocala Women’s Center.

abortion website

As one would expect from a company using deceptive advertising practices, the home page shown above is also full of deception. First and foremost is the quote at the top right, “I was relieved it could be so simple.” The “it” being referred to is the killing of an unborn child by taking a pill. This is not simple, and can result in severe injury and even the mother’s death, not to mention the lifelong psychological trauma that haunts many mothers when they realize what they have done.

Another deceptive tactic is in the paragraph I circled in red. This paragraph clumsily attempts to say something about abortion numbers not dropping quickly because poor women do not have access to family planning centers, education and birth control. However, the main reason women have an abortion has nothing to do with poverty. As the page shows, the rate of unborn child-killing may be higher for women below the poverty level (112 vs. 29 per 1000 women), but just because the rate is higher, does not mean the number of abortions is higher, or that poverty is the main reason. 93% of all abortions are the result of social reasons, meaning the child is inconvenient or unwanted (http://www.abortionno.org/Resources/fastfacts.html). This paragraph is irrational and deceptive. If the “incidence of becoming pregnant has drastically increased” in America, while the number of unborn child-killings has decreased, isn’t that a good thing?

Yes, a decrease in the legal murder of unborn children, coupled with an increase in the incidence of pregnancy is a good thing. What is probably happening is not a lack of education, but rather more and more people are being educated about the real definition of abortion, and they are choosing to keep their babies. And if that baby is truly inconvenient or unwanted, there are plenty of people who would be willing to adopt the child. No child deserves to be labeled as “unintended” or “unwanted”, and if a child receives that label, he or she doesn’t deserve to be killed for it.

Abortion is murder, and murder is sin. In the documentary Killing Girls, about late-term abortion in Russia, the film follows the life of a young Russian teenager whose situation is not very different than many other girls around the world who get abortions. The girl lives with her mother, and the father left long ago. So the “trail of sin” begins with an unfaithful father. From there, the girl sins by getting drunk, then sins by committing adultery, gets pregnant, almost sins by carrying out the child-killing procedure, but decides not to at the last minute. She has a healthy baby girl who brings a new sense of life and purpose to both her and her mother. About a year later, she commits adultery again, gets pregnant again, but decides to kill the second baby, even though there are plenty of people in Russia and elsewhere willing to adopt a child. Her excuses are typical, the baby would be too expensive, too much of a burden, etc., which may be true, but ultimately these are selfish reasons. Other girls interviewed on the documentary have abortions for excuses ranging from rape to concerns about how having a baby will affect their figure.

The sin of murdering unborn children is having a negative effect on our world. One of the biggest impacts is economic, as highlighted in this article from lifesitenews. As the article points out, we do not have a “debt crisis”, we have a “death crisis”. To have a successful business, people must buy your goods. To have a successful country, people must support the government with at least some level of taxation. The current worldwide “war on unborn children”, resulting in the legalized murder of about 42 million children per year, is resulting in fewer people to buy things, and fewer people to pay taxes. Businesses close and governments collapse because there are not enough people to buy the goods produced and not enough workers to pay the taxes required to maintain even a basic government infrastructure. One example is Greece, which had a major economic crisis come to light in 2010. According to the CIA World Factbook, Greece has a fertility rate of 1.37, ranking it at #203 out of 224 countries. In order to maintain a stable population, a country needs a fertility rate of 2.0 children born per woman. Greece, and many other countries, have had fertility rates below 2.o for years, and the effects of having too few children are becoming more and more visible. Killing unborn children has been legal in Greece since 1984.

Abortion is the murder of an innocent child. It is bad for the mother and father, bad for the community, bad for the city, bad for the nation, and bad for the world. Until His kingdom comes on Earth, as it is in Heaven (Luke 11: 2-3), we humans are going to keep on sinning, but people don’t have to try to cover up the sins of adultery or selfishness with the sin of murdering an innocent, unborn child. In America, we declared independence because our founding fathers believed it was a self-evident truth that all humans are endowed by their Creator with the right to live, the right to liberty, and the right to pursue happiness. Every child, born and unborn, deserves these rights, and it should be against the law to deny a child such rights.

Last week, a student who uses our DIVE CDs emailed me to verify an error in problem number 24 of Saxon Calculus, 2nd edition, Problem Set 63. Although this error did not actually affect the final answer, it was an error nonetheless, and was missed by the editors of the Saxon Solutions Manual. EVERY curriculum has a few errors, including the products my company, DIVE LLC, produces, so the point of this article is not to ridicule Saxon Publishers for their errors. The point is to talk about educating children. If you visit the DIVE website or read a DIVE catalog, you will quickly understand that I think it is time to “raise the standard” in K-12 education, and this includes completing calculus in high school. I realize not every student can accomplish this goal, but there is nothing wrong with at least making that the goal with the possibility of falling short.

So what is the big deal about this student who found an error? Well, he’s 12 years old. And he is home educated. By his parents. Obviously, this boy has an aptitude for math that exceeds that of most 12-year olds and even many adults, but the fact remains that he is almost halfway-through a course that is the equivalent of college Calculus I. And if there are 12-year olds out there who can do calculus, there are even more 13, 14, 15+ year olds who can do calculus.

Error in Saxon Calculus 2nd edition solutions manual. Okay, so the handwriting is not super-neat, but give the kid a break, he's 12!

Most people today realize the potentials of homeschooling, but strangely enough, there are still those who want to eliminate it. According to an article in the November/December 2010 issue of The Home School Court Report, American homeschooling is entering a “Third Wave” of persecution. The first wave had to do with education. Could parents really educate children at home? While there are some parents who do a really bad job at educating, the same can be said of many government and private schools. And in today’s world, 12-year olds doing calculus would be a real surprise in a government school and most private schools, but not in a home school.

The second wave of persecution was about the issue of socialization, but the evidence now weighs heavily against that idea, too. While the first two waves of persecution were based on false premises, the third wave of persecution is, according to HSLDA Chairman Mike Farris, essentially correct. According to Farris “Christian homeschooling parents are effectively transmitting values to their children that the elitists believe are dangerous to the well-being of both these very children and society as a whole.”

I believe God gave us the ability to do mathematics so that we could better understand His creation. I believe we should study math and science so we can know Him better and as Christians, act on the faith He gave us and get out there and be excellent at doing the work He has already prepared for us (Ephesians 2:8-10). I also believe we can study math and science without acknowledging its Author, but to do so, we miss half the story. There are those in society who actually think it would be better for us all if Christian 12-year old boys with above-average math aptitude were taken from their loving parents, placed in a large group, and taught a below-aptitude, godless curriculum. The boys would not learn that they were created in the image of the greatest Creator, and are therefore creative, too. Instead, their creativity will be stifled, supposedly so they fit in better with the group, and they will instead learn they were not created by a loving Father, but rather evolved from a meaningless pile of goo in a mysterious land that existed before history. And the people who promote such ideas are referred to as the “elites”? Really?

The third wave of homeschool persecution is nothing more than misguided “elitists” wanting to replace Christian religion, where we are taught to love the sinner but hate the sin, with a secular fundamentalist religion that teaches to hate some sinners and love many sins. The first two waves taught us home schoolers to surf. Skills learned in the past will help us surf this bigger, badder, third wave, too, and ensure the current freedoms American parents have to teach calculus to their math-gifted 12-year olds. But we won’t keep these freedoms if we sit around and do nothing, so get out there and surf!